(Adds update tag to headline field)
* Ukrainian government warplanes carry out airstrikes
* One man killed by shrapnel from shell
* Rebels declared 'people's republics' in eastern Ukraine
By Sabina Zawadzki and Gabriela Baczynska
DONETSK, Ukraine, May 26 (Reuters) - Ukrainian government warplanes carried out airstrikes against pro-Russian rebels who seized Donetsk international airport on Monday, triggering a battle in and around the complex in a show of force by both sides after the election of a new president.
Reuters journalists saw black smoke billowing from the area of the airport after repeated explosions and gunfire, while jets roared overhead. A security official also said paratroops had landed in one of the fiercest clashes since violence broke out in Russian-speaking eastern Ukraine some two months ago.
Loud thuds sounded from the area but after four to five hours Ukrainian soldiers appeared to have pushed back at least some of the separatists towards the town, though it was not clear if the army had recovered control of the airport.
There was no full account from either side of casualties. But the press service of the regional administration said one man had been killed and two injured from shrapnel of a shell which struck a part of the railway station serving the airport.
Saying that a deadline had passed at 1 p.m. (1000 GMT) for separatist militants to lay down their arms, a spokesman for the Ukrainian joint forces' security operation in the region said two Sukhoi Su-25 jets had carried out strafing runs, firing warning shots around Sergei Prokofiev International Airport.
"In reply, the guerrillas opened fire at random from all types of weapon," he said, before a MiG-29 jet also took part.
A second spokesman, Vladislav Seleznyov, said: "A MiG-29 carried out an airstrike on the area where the terrorists were concentrated."
"CLEANING UP THE AREA"
The first spokesman said the militants had then spread out across the territory of the airport, whose state-of-the-art main terminal was built only for the 2012 European soccer championships held in Ukraine. "Right now at the airport, paratroops have landed and are cleaning up the area."
He appeared to mean that troops had landed by helicopter.
The action, after an overnight move by rebels of the Donetsk People's Republic to take over the airport, looked like a forceful first act by Petro Poroshenko, the billionaire who swept to the presidency on Sunday. He said he would not treat with "terrorists", despite calls from Russia for Kiev not to step up its halting military operations in the east of Ukraine.
Seleznyov said a helicopter had been in action to destroy an anti-aircraft battery being used by rebels. He denied a report from the rebel side that a helicopter had been brought down.
He later said the rebels had received reinforcements - though he did not say from whom - adding that fighting had now entered "an active phase. Government forces, he said, aimed to clear rebels from the airport. He declined to give casualties.
Warnings were issued by radio for inhabitants of residential blocks in districts closest to the airport to stay inside and, if shooting occurred, to take refuge in communal cellars.
A local news website published a photograph of three men in camouflage with a rapid-fire grenade launcher pointing skywards on what appeared to be the modern glass and concrete roof of the airport terminal. It said they were separatist rebels.
One Reuters journalist, leaving a vantage point in one block, saw armed rebels gathered on lower floors of the building while Ukrainian soldiers moved around on streets nearby.
Earlier, Reuters journalists saw three trucks carrying armed militants heading towards the airport and also spotted at least four armed men positioned on the airport roof. Representatives of the Donetsk People's Republic said they had taken over the airport and wanted to clear it of government forces there.
Donetsk, a city of a million that is effectively the capital of the industrial Donbass region, is largely in the hands of pro-Russian separatists who prevented local people taking part in the presidential election on Sunday.
Donetsk airport authorities announced its closure to flights early on Monday after the separatists came to the facility to demand the withdrawal from the area of Ukrainian forces, who have been policing the perimeter.
The pro-Moscow rebels have declared autonomous "people's republics" in the regions of Donetsk and Luhansk following makeshift referendums on May 11. They say the two regions are no longer part of Ukraine.
(Additional reporting by Yannis Behrakis in Donetsk and Pavel Polityuk in Kiev; writing by Alastair Macdonald and Richard Balmforth; Editing by Gareth Jones)